Chenyang An

Chenyang An, Xiaoqian Xu

We establish explicit lower bounds for advection-diffusion equations in three settings: a polynomial $\dot H^{-1}$ bound for inviscid shears with $u\in L^\infty_t W^{1,1}_y$, a uniform positive lower bound on the mixing scale for diffusive shears, and an exponential $L^2$ bound for rapidly oscillating time-periodic flows. All constants are explicit in the data. The proofs were generated entirely by a multi-agent math proving system, QED, without expert human intervention, serving as a test of AI's capability to produce rigorous mathematics.

Chenyang An, Qihao Ye, Minghao Pan et al.

We explore a central question in AI for mathematics: can AI systems produce original, nontrivial proofs for open research problems? Despite strong benchmark performance, producing genuinely novel proofs remains an outstanding challenge for LLMs. Through systematic experiments with frontier LLMs on research-level proof tasks, we identify seven failure modes that prevent reliable proof generation, including context contamination, citation hallucination, hand-waving on key steps and misallocation of proof effort, unstable proof plans, unfocused verification, problem modification and single-model bottleneck. We argue that the gap between benchmark success and research-level proving is primarily one of system design, due to those failure modes. We present QED, an open-source multi-agent proof system in which each architectural decision directly addresses a specific failure mode. Evaluated on five open problems in applied analysis and PDEs contributed by domain experts, QED produces correct proofs for three problems, each verified by the contributing experts as original and nontrivial. QED is released as open-source software at https://github.com/proofQED/QED.

Chenyang An, Zhibo Chen, Qihao Ye et al.

Recent advances in Automated Theorem Proving have shown the effectiveness of leveraging a (large) language model that generates tactics (i.e. proof steps) to search through proof states. The current model, while trained solely on successful proof paths, faces a discrepancy at the inference stage, as it must sample and try various tactics at each proof state until finding success, unlike its training which does not incorporate learning from failed attempts. Intuitively, a tactic that leads to a failed search path would indicate that similar tactics should receive less attention during the following trials. In this paper, we demonstrate the benefit of training models that additionally learn from failed search paths. Facing the lack of such trial-and-error data in existing open-source theorem-proving datasets, we curate a dataset on intuitionistic propositional logic theorems and formalize it in Lean, such that we can reliably check the correctness of proofs. We compare our model trained on relatively short trial-and-error information (TrialMaster) with models trained only on the correct paths and discover that the former solves more unseen theorems with lower trial searches.

Longfei Yun, Chenyang An, Zilong Wang et al.

Instruction-tuned large language models (LLMs) employ structured templates, such as role markers and special tokens, to enforce format consistency during inference. However, we identify a critical limitation of such formatting: it induces a phenomenon we term diversity collapse, where the model generates semantically similar outputs for open-ended inputs, undermining creativity and variability. We systematically evaluate this effect across tasks like story completion and free-form generation, finding that (1) diversity collapse persists even under high-temperature sampling, and (2) structural tokens in templates significantly constrain the model's output space. To contextualize these findings, we fine-tune the same model using a range of structured prompts and then evaluate them across three axes: downstream task performance, alignment behavior, and output diversity. Our analysis shows that format consistency between fine-tuning and inference is crucial for structure-sensitive tasks (e.g., GSM8K, IFEval), but has marginal influence on knowledge-heavy tasks (e.g., MMLU, WebQuestions). In contrast, output diversity is primarily governed by the presence or absence of structural tokens, with minimal formatting yielding the most diverse outputs. These findings reveal that current prompting conventions, while beneficial for alignment, may inadvertently suppress output diversity, underscoring the need for diversity-aware prompt design and instruction tuning.

Letian Peng, Chenyang An, Shibo Hao et al.

The generalization of language models (LMs) is undergoing active debates, contrasting their potential for general intelligence with their struggles with basic knowledge composition (e.g., reverse/transition curse). This paper uncovers the phenomenon of linear correlations in LMs during knowledge composition. For explanation, there exists a linear transformation between certain related knowledge that maps the next token prediction logits from one prompt to another, e.g., "X lives in the city of" $\rightarrow$ "X lives in the country of" for every given X. This mirrors the linearity in human knowledge composition, such as Paris $\rightarrow$ France. Our findings indicate that the linear transformation is resilient to large-scale fine-tuning, generalizing updated knowledge when aligned with real-world relationships, but causing hallucinations when it deviates. Empirical results suggest that linear correlation can serve as a potential identifier of LM's generalization. Finally, we show such linear correlations can be learned with a single feedforward network and pre-trained vocabulary representations, indicating LM generalization heavily relies on the latter.

Ali Abbasi, Shima Imani, Chenyang An et al.

With the rapid scaling of neural networks, data storage and communication demands have intensified. Dataset distillation has emerged as a promising solution, condensing information from extensive datasets into a compact set of synthetic samples by solving a bilevel optimization problem. However, current methods face challenges in computational efficiency, particularly with high-resolution data and complex architectures. Recently, knowledge-distillation-based dataset condensation approaches have made this process more computationally feasible. Yet, with the recent developments of generative foundation models, there is now an opportunity to achieve even greater compression, enhance the quality of distilled data, and introduce valuable diversity into the data representation. In this work, we propose a two-stage solution. First, we compress the dataset by selecting only the most informative patches to form a coreset. Next, we leverage a generative foundation model to dynamically expand this compressed set in real-time, enhancing the resolution of these patches and introducing controlled variability to the coreset. Our extensive experiments demonstrate the robustness and efficiency of our approach across a range of dataset distillation benchmarks. We demonstrate a significant improvement of over 10% compared to the state-of-the-art on several large-scale dataset distillation benchmarks. The code will be released soon.

Chenyang An, Shima Imani, Feng Yao et al.

In the field of large language model (LLM)-based proof generation, despite extensive training on large datasets such as ArXiv, LLMs still exhibit only modest performance on proving tasks of moderate difficulty. We believe that this is partly due to the widespread presence of suboptimal ordering within the data for each proof used in training. For example, published proofs often follow a purely logical order, where each step logically proceeds from the previous steps based on the deductive rules. This order is designed to facilitate the verification of the proof's soundness, rather than to help people and models learn the discovery process of the proof. In proof generation, we argue that the optimal order for one training data sample occurs when the relevant intermediate supervision for a particular proof step in the proof is always positioned to the left of that proof step. We call such order the intuitively sequential order. We validate our claims using two tasks: intuitionistic propositional logic theorem-proving and digit multiplication. Our experiments verify the order effect and provide support for our explanations. We demonstrate that training is most effective when the proof is in the intuitively sequential order. Moreover, the order effect and the performance gap between models trained on different data orders can be substantial -- with an 11 percent improvement in proof success rate observed in the propositional logic theorem-proving task, between models trained on the optimal order compared to the worst order. Lastly, we define a common type of order issue in advanced math proofs and find that 17.3 percent of theorems with nontrivial proofs in the first two chapters of a widely used graduate-level mathematics textbook suffer from this issue. A detailed list of those proofs is provided in the appendix.